The axiom of choice

Content created by Egbert Rijke, Fredrik Bakke and Jonathan Prieto-Cubides.

Created on 2022-02-16.
Last modified on 2024-02-06.

module foundation.axiom-of-choice where
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.functoriality-propositional-truncation
open import foundation.postcomposition-functions
open import foundation.projective-types
open import foundation.propositional-truncations
open import foundation.sections
open import foundation.split-surjective-maps
open import foundation.surjective-maps
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.identity-types
open import foundation-core.precomposition-functions
open import foundation-core.sets


The axiom of choice asserts that for every family of inhabited types indexed by a set, the type of sections of that family is inhabited.


The axiom of choice restricted to sets

AC-Set : (l1 l2 : Level)  UU (lsuc l1  lsuc l2)
AC-Set l1 l2 =
  (A : Set l1) (B : type-Set A  Set l2) 
  ((x : type-Set A)  type-trunc-Prop (type-Set (B x))) 
  type-trunc-Prop ((x : type-Set A)  type-Set (B x))

The axiom of choice

AC-0 : (l1 l2 : Level)  UU (lsuc l1  lsuc l2)
AC-0 l1 l2 =
  (A : Set l1) (B : type-Set A  UU l2) 
  ((x : type-Set A)  type-trunc-Prop (B x)) 
  type-trunc-Prop ((x : type-Set A)  B x)


Every type is set-projective if and only if the axiom of choice holds

is-set-projective-AC-0 :
  {l1 l2 l3 : Level}  AC-0 l2 (l1  l2) 
  (X : UU l3)  is-set-projective l1 l2 X
is-set-projective-AC-0 ac X A B f h =
    ( ( map-Σ
        ( λ g  ((map-surjection f)  g)  h)
        ( precomp h A)
        ( λ s H  htpy-postcomp X H h)) 
      ( section-is-split-surjective (map-surjection f)))
    ( ac B (fiber (map-surjection f)) (is-surjective-map-surjection f))

AC-0-is-set-projective :
  {l1 l2 : Level} 
  ({l : Level} (X : UU l)  is-set-projective (l1  l2) l1 X) 
  AC-0 l1 l2
AC-0-is-set-projective H A B K =
    ( map-equiv (equiv-Π-section-pr1 {B = B})  tot  g  htpy-eq))
    ( H ( type-Set A)
        ( Σ (type-Set A) B)
        ( A)
        ( pr1 ,  a  map-trunc-Prop (map-inv-fiber-pr1 B a) (K a)))
        ( id))

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